Weakly Precomplete Equivalence Relations in the Ershov Hierarchy

Abstract

We study the computable reducibility ≤c for equivalence relations in the Ershov hierarchy. For an arbitrary notation a for a nonzero computable ordinal, it is stated that there exist a $$ {\varPi}_a^{-1} $$ -universal equivalence relation and a weakly precomplete $$ {\varSigma}_a^{-1} $$ - universal equivalence relation. We prove that for any $$ {\varSigma}_a^{-1} $$ equivalence relation E, there is a weakly precomplete $$ {\varSigma}_a^{-1} $$ equivalence relation F such that E ≤cF. For finite levels $$ {\varSigma}_m^{-1} $$ in the Ershov hierarchy at which m = 4k +1 or m = 4k +2, it is shown that there exist infinitely many ≤c-degrees containing weakly precomplete, proper $$ {\varSigma}_m^{-1} $$ equivalence relations.